A New Mathematical Dynamic Model for HVAC System Components Based on Matlab Simulink

International Journal of Innovative Technology and Exploring Engineering (IJITEE)

ISSN: 2278-3075, Volume-1, Issue-2, July 2012

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Abstract— In this paper, a new and complete mathematical

dynamic model of HVAC (Heating, Ventilating, and Air

Conditioning) components such as heating/cooling coil,

humidifier, mixing box, ducts and sensors is described. All of

these components are proposed and simulated in

Matlab/Simulink platform. The proposed model is presented in

terms of energy mass balance equations for each HVAC

component. We have considered two control loop for this model,

namely, temperature control loop and humidity ratio control

loop. The proposed model is a full dynamic model of HVAC

system that includes least approximations and assumes.

Index Terms— HVAC system, HVAC components,

Matlab/Simulink, HVAC model

I. INTRODUCTION

Initially, the most important issue of HVAC (Heating,

ventilating and Air-Conditioning) systems factories was to

maintain the zone conditions in predefined values related to

occupants' thermal comfort. However, with start of energy

crisis, the amount of energy consumption of these

equipments became important. In order to, evaluate these two

options, the designer have been started to modeling and

design of new type of HVAC system components.

In this paper, some study has been focused on HVAC

modeling. For instance, Wang et al. [1] have used heat

transfer and energy balance principles to identify a linear

model to represent a non-linear model of a cooling coil.

Clarke [2] has presented a transient model for a HVAC

system for some components such as humidifier and mixing

box, but no specific model for cooling/heating coils was

given. Stoecker [3] provided a model for cooling coil with

empirical parameters under the assumptions of constant air

flow and water flow. Braun and Rabehi [4,5] presented two

cooling coil models that were too complex and iterative

computations. Many researchers studied HVAC dynamic

models using empirical or theoretical methods. Underwood

and Crawford [6] developed a nonlinear model of a heat

exchanger loop. Nassif et al. [7] presented a validation of the

cooling coil, mixing box and a fan for a VAV (Variable Air

Volume) system.

An issue related to HVAC systems modeling, is the use of an

Manuscript received on July, 2012

Ahmad Parvaresh, Department of electrical engineering, Shahid Bahonar

university of Kerman, Kerman, Iran.

S. M. Mohammadi, Department of electrical engineering, Shahid Bahonar

university of Kerman, Kerman, Iran.

Ali Parvaresh, Department of electrical engineering, Shahid Rajaee

Teacher Training University, Tehran, Iran.

appropriate simulation tool. We can categorize these

modeling tools on three sections.

First, tools that used for pipe/duct sizing, as AFT Fathom,

DOLPHIN, DUCTSIZE, EES (Energy Equation Solver) and

PHYTON.

Second, tools that used for energy performance analysis, as

Carrier HAP, TRANSYS, and HVACSIM+.

Third, tools that used for control and modeling analysis of

these systems, as MATLAB and Energy Plus.

Advantage and disadvantage of these tools have mentioned

in [8]. With these programs, the heating, cooling, lighting,

ventilation and other energy related flows in a building can

be simulated. Lebrun [9] presented a simulation of a HVAC

system by using EES.

In these groups the reason for choosing Simulink, which is

part of the Matlab package, was a larger degree of flexibility,

modular structure, and transparency of the models and ease

of use in the modeling process. The modular structure in

Simulink makes it easier to maintain an overview of the

models, and new models can just as easily be added to the

pool of existing models [8]. Recently, some

Matlab/Simulink-based simulations of HVAC systems have

been developed. Some examples of these studies are Reiderer

et al. [10], Mendes et al. [11], Dion et al. [12], Huang and

Lam [13], Ghumari et al. [14], Oliveira et al. [15].

In this paper we have focused on the modeling of HVAC

system components and simulation of the presented model.

The HVAC system components that modeled are the zone,

the cooling coil, the heating coil, the humidifier, the mixing

box, the ducts, and the sensor models. This paper is

organized as follows: Section 2 presents an introductory

explanation of the HVAC system and its components models.

Section 3 covers details of the new proposed model. In

sections 4 and 5 we have simulated presented model in

Matlab/Simulink and discussion about results, respectively.

Finally, in section 6 we have mentioned conclusion.

II. MODEL OF HVAC SYSTEM COMPONENTS

In recent years, several numerical simulation models for

HVAC systems performance were developed. Because of

complexity of these systems a complete theoretical approach

of formulating the model is too difficult. In order to, achieve

full dynamic model of HVAC system we have to reach all of

the important models of components. The major components

considered in the system model can be divided in two groups,

which are the zone model and Components of HVAC system.

First we propose a model for the zone and then for all of

A new mathematical dynamic model for HVAC

system components based on Matlab/Simulink

Ahmad Parvaresh, Seyed Mohammad Ali Mohammadi, Ali Parvaresh

A new mathematical dynamic model for HVAC system components based on Matlab/Simulink

2

components that are in HVAC system.

Figure 1. View of energy and mass balance equation that used for the zone modeling

A. The Zone Model

One of the most important sections of HVAC system

model is the zone modeling. All of components in HVAC

systems work to the zone achieve to optimal conditions. In

previous works, because of simplify and approximation in

presented models, they are far from their real performance.

In our proposed model we have considered effect of

uncontrolled input as people, lights and so on, and effect of

the north wall, south wall, east wall, west wall, floor and

inner roof on the zone temperature. In this model we have

considered eight state variables: the zone temperature (Tz),

inner wall temperature (Tws,Twn,Twe,Tww,Tf,Tr), and the zone

humidity ratio (Wz). The pressure of air is assumed to be

constant and air in the zone in fully mixed.

With these descriptions, energy and mass balance

equations are (as shown in Fig. 1)

( - )

( - ) ( - )

( - ) ( - )

( - ) ( - )

dTzC T Tf C paasaz sa zdt

H A T T H A T Tws ws ws z wn wn wn z

H A T T H A T Twe we we z ww ww ww z

H A T T H A T T Qr r r z f f f z

(1)

( )+ ( )wsws ws ws z ws ws ws o ws

dTC H A T T H A T T

dt (2)

( )+ ( )wnwn wn wn z wn wn wn o wn

dTC H A T T H A T T

dt (3)

( )+ ( )wewe we we z we we we o we

dTC H A T T H A T T

dt (4)

( )+ ( )wwww ww ww z ww ww ww o ww

dTC H A T T H A T T

dt (5)

( )+ ( )wnr r r z r r r o r

dTC H A T T H A T T

dt (6)

( )f

f f f z f

dTC H A T T

dt (7)

( )zz s s z

dWV f W W

dt (8)

With taking the Laplace transform of Eqs. (1)- (8) and

rearrange we have

2

2 2 2

2 2

( ) ( )[ ( ) [ ( )

+ ( ) ( ) ( )

+ ( )]( ( ) ( )) ( ) ( ) ]

z z s ws

wn we ww

r z o f f

T s G s T s G s

G s G s G s

G s T s T s G s T s Q

(9)

( ) ( ) ( )z wz sW s G s W s QQ (10)

Where

+

ws ws wn wn

we we ww ww r r f f

f C H A H Asa a pa

H A H A H A H A (11)

, , ,

, , , .

ws ws wn wn

we we ww ww r r f f

f C H A H Asa a pa

H A H A H A H A (12)

1 1( ) , ( ) ,

2 2

1 1( ) , ( ) .

2 2

wsws

wn wewn we

G s G sz C s C sz

G s G sC s C s

(13)

1 1( ) , ( ) ,

2

1( ) , ( ) .

ww rww r

f wzf

G s G sC s C s

fsG s G sC s V s fz s

(14)

Figure 2. Graphical view of the zone

Fig. 2 shows a graphical view of the zone that modeled.

B. The Cooling Coil Model

The most important component of every HVAC system is

cooling coil. The increased emphasis on variable operation of

HVAC equipment warrants a greater understanding of the

dynamic behavior of cooling coils. Due to importance of this

component, many studies have done as [16], [17], and [18].

In Fig. 3 a view of cooling coil is shown.



3

Figure 3. A view of cooling coil In air side

,( ) ( ), ,

dTa outT T T Ta t a out a indt

(15)

Where

, , ,

,

M c T M c Ta p a a in w w a outTa M c M ca p a w w

(16)

And

,,

h A Mt s ov o a

c A ALv c

(17)

We can rewrite Eq. 16 as follow

( ) ( ) ( ), ,T s T s T sa a in a out (18)

That in which

,, .

, ,

M c M ca p a w w

M c M c M c M ca p a w w a p a w w

(19)

By taking Laplace transform Eq. 15

( ) ( ) ( ( ) ( )) ( ),s T s T s T s T sa a t a in (20)

We obtain ( )T st by substituting Eq. 18 in Eq. 20

( ) ( ) ( ), ,s

T s T s T st a out a in (21)

In water side

( ) ( )dTwr T T T Tt w ws wrdt

(22)

Where

,

,

M c T M c Ta p a wr w w wsTw M c M ca p a w w

(23)

And

, it it w

w w w c

h A M

m c m L (24)

With rewriting Eq. 23 we have

( ) ( ) ( )w wr wsT s T s T s (25)

By taking Laplace transform Eq. 22

( ) ( ) ( ) ( ) ( )s T s T s T s T swr t w ws (26)

We obtain ( )T st by substituting Eq. 25 in Eq. 26

( ) ( ) ( )t wr ws

sT s T s T s (27)

By equating right-hand sides of Eq. 21 and Eq. 27

( ) ( ),

( ) ( )

sT s T sa a in

sT s T swr ws

(28)

With simplify Eq. 28 we can obtain ( )T sa in terms

of ( )T swr , ( )T sws , and ( ),T sa in

( ) ( )( ) ( ) ( )

( ) ( )

( ),( )

sT s T s T sa wr wss s

T sa ins

(29)

C. The Heating Coil Model

In HVAC systems heating coils are placed in the airstream

to regulate the temperature of the air delivered to the space.

During the heating mode, problems can occur if the hot water

temperature in the heating coil has been set too low in an

attempt to reduce energy consumption. If enough outdoor air

to provide sufficient ventilation is brought in, that air may

not be heated sufficiently to maintain thermal comfort or, in

order to adequately condition the outdoor air, the amount of

outdoor air may be reduced so that there is insufficient

outdoor air to meet ventilation needs.

In proposed model for heating coil we considered effect of

input/output water temperature through the coil, output

temperature of the zone and input air from mixing box on

output air temperature of coil.


4

( )

( )( )

( )

dT co fC C T Twi woah pwwswdt

UA T To coa

f C T Tm copaasa

(30)

[ ] ( )( )

[ ( ) ( )]

( ) ( )( )

cos T sf UAC Cah paasa a

s sf C T Twi wopwwsw

s sUA f CT To mpaaa sa

(31)

( ){ [ ( ) ( )] ( ) ( )}s s s s sGT T T T Tco wi wo o ms (32)

Where

( ) , , ( ) ,

1( ) .

UA f UAf C Cpa pwsa a a sw w a

G ss C sah

(33)

( ) ( ) ( )V f W s f W sah sa co sa m (34)

( ) ( ) ( )W s G s W ss ws si (35)

( )fsaG sws V s fah sa

(36)

D. The Humidifier Model

The measurement and control of moisture in the air is an

important phase of air conditioning [19]. In some references

has been presented model for humidifier as [19] and [20].

( ) ( )dT h fC C T T T Tsi h o hh pasa hdt

(37)

( )( )

dW h thV f W Wh sa si hdt a

(38)

E. The Mixing Box Model

A mixing box is the section of an air handling unit used to

mix the return air flow with the outside air flow. It consists of

three sets of dampers whose operation is coordinated to

control the fraction of the outside air in the supply air while

maintaining the supply airflow rate approximately constant.

In Fig. 4 structure of air flow in the mixing box unit is shown.

m C T m C T m C Tr pa r o pa o m pa m (39)

m m mr o m (40)

m T m Tr r o oTm m mr o

(41)

m W m Wr r o oWm m mr o

(42)

Figure 4. The mixing box structure

F. The Duct Model

The ducts are used in HVAC systems to deliver and

remove air. Air ducts are one method of ensuring acceptable

indoor air quality as well as thermal comfort.

( )( )

h h m CdT i o a pout T Tin outdt h M Ci c c

(43)

( ) ( ) ( )T s G s T so duct in (44)

( )( ) ,

h h m Ci o a pG sduct s h M Ci c c

(45)

,( ) ( ), ,1,

s tempT s T ss temp m tempss temp

(46)

G. The Sensors Model

The sensors that are used in HVAC systems have different

types as temperature sensor, Humidity sensor, pressure

sensor, and flow sensor. In many studies different models

have been presented to these sensors. For instance, Wang

[17] by using of time constant manner considered a first

order differential equation for temperature and humidity

sensor.

Note that for achieve more exact model we have to

consider different time constant. In proposed model, we have

considered a first order differential equation with time

constant 1 for temperature sensor and time constant 2 for

humidity sensor.

Fig. 6 and Fig. 7 show changes in temperature and relative

humidity over the period 1th-31th march 2012 in Kerman,

respectively.

III. SIMULATION RESULTS OF PROPOSED MODEL

In order to simulate suggested system, the average

temperature and relative humidity of Kerman are compared

for mentioned period and are considered as two inputs for

mixing box subsystem. In this regards, we consider two

control loops so as to control temperature and relative

humidity.

,( ) ( ), ,1,

s humT s T ss hum m humss hum

(47)



5

Figure 5. Proposed model for HVAC system

IV. PROPOSED MODEL

The proposed model for a HVAC system is presented in

Fig. 5. This figure shows all components as subsystem in

simulink platform. Moreover, the components arrangement

is based on their real models.

We applied output temperature and relative humidity to

model, then simulated it. We considered two PID controllers

for control the loops.

Fig. 8 indicates the temperature-time curve of our

suggested model. From this figure, the zone temperature

tracks the set point very well, at time 40s arrive to desired

value, and speed of convergence is high.

The relative humidity-time curve is presented in Fig. 9 for

our suggested model. According to figures, it is revealed that

suggested model track the set point with less time constant

and high speed.

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

Days (1-31 march2012)

Te

me

ratu

re (

C)

Figure 6. Changes of temperature over the period 1th-31th

march 2012

0 5 10 15 20 25 30 350

10

20

30

40

50

60

70

80

Days(1-31 march2012)

Hu

mid

ity

(%

)

Figure 7. Changes of relative humidity over the period

1th-31th march 2012

0 20 40 60 80 100 120 140 160 180 20017

18

19

20

21

22

23

24

Time(S)

Te

mp

era

ture

(C)

Figure 8. The zone temperature- time curve of proposed

model


6

0 50 100 150 200

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

Time(S)

Rela

tive h

um

idit

y(/

100)

Figure 9. The humidity ratio- time curve of proposed model

V. CONCLUSION

In this paper we have presented a dynamic model for

HVAC system components as the zone, cooling coil, heating

coil, humidifier, mixing box, the ducts and the sensors. After

derive mathematical models of components we analyzed

simulation system of the complete HVAC system in

Matlab/Simulink platform. In this models, which have been

presented up to now, several simplifier assumptions such as

effect of floor and roof on zone temperature, have been

considered. But, we present a model, which is close to real

model. Indeed, we present a complete mathematical dynamic

model. The simulation results demonstrated that our

suggested model for HVAC system components can be work

very well.

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d’un système CVCA existent”. Vie Col. Interuniversitaire

Franco-Québécois, Canada.

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engineering equation solver”, 7th IBPSA Conference, Brazil.

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[20] B.Tashtoush, M. Molhim, M. Al-Rousan, “Dynamic model of an HVAC

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Ahmad Parvaresh received B.S degree in Electrical

Engineering from Tafresh University, Tafresh, Iran, in

2010. Currently, he is a M.Sc. degree in Shahid Bahonar

University of Kerman, Iran. His interests include

industrial automation, adaptive control, and control of

HVAC systems.

Seyed Mohammad Ali Mohammadi received his B.S.

degree in Electrical Engineering from the Sistan and

Baluchistan University, Zahedan, Iran, in 1997, M.Sc.,

degree from the Esfahan University of technology,

Esfahan, Iran, in 2002 and PhD degree in Applied

Mathematics in Control Systems from S.B.U.K, Kerman,

Iran in 2009. His research interests are in fuzzy systems,

artificial intelligence, instrumentation and control

systems.

Ali Parvaresh received his B.S degree in electrical

Engineering from the Shahid Rajaee Teacher Training

University, Teharan, Iran, in 2006, M.Sc. degree from

the Shahid Rajaee Teacher Training University, Tehran,

Iran, in 2008. His research interests are in power systems

analysis, dynamic power systems.

A New Mathematical Dynamic Model for HVAC System Components Based on Matlab Simulink

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